An Intuitive Guide To Exponential Functions & e

Hi great article

i have a question. In the article you say this:

What if we grow at 50% annually, instead of 100%?
If we pick n=50, we can split our growth into 50 chunks of 1% interest:

(1+.50/50)^50=(1+0.01)^100/2=e^1/2

I understand this, but why did you use n= 50, could you use n=10 in this case would it still work ? thanks

hi great article,

you say at the end of the video and in the article that e can approximate jagged systems that don’t grow smoothly. Can you give an example of this because I don’t see it.
Thanks

how can I apply exponential function to measure/analyze our own growth ?

Hi Khalid,

Very informative post. But I think the concepts tend to evaporate over time hence there must be some place to apply these concepts so that it sticks there for long time. Do you know something that may help us to achieve this thing.

PS: I am a programmer to so, its very difficult to go back to school books.

Thanks!

Thankyou for the great article - i’m learning about cell survival curves , the survival fraction of cells after a radiation dose. It is describes as e^-(ax + bx^2) . Trying to get an intuitive feel to this :slight_smile: is there a way to get the head around e^-(x^2) ? or how about the general e^(x^n) ??

@dg: Great question. When dealing with continuous growth, we can mix and match rate and time as we please. Here’s why.

Let’s say we have 1 year of 30% continuous growth. That means each individual dollar intends create interest of 30% when all is said and done. Along the way, its interest will start growing, but the original dollar doesn’t know! The 30% growth (from a continuous case) means “An individual dollar, Mr. Blue, knows he will create a Mr. Green that is 30% as large at the end of the period. What Mr. Green happened to be doing along the way is none of Mr. Blue’s business.”

So, with this model, it doesn’t matter if Mr. Blue builds up Mr. Green in one year of 30%, or in 30 years of 1% progress each year. At the end, Mr. Blue has made a Mr. Green who is 30% as large as he is.

Clearly, it’s better for us, the investors, to get the growth in 1 year vs. 30, but either approach results in the same total amount (e^{30% * 1} = e^{1% * 30} = e^{.30} = 1.35. Either way, have a total growth from 1 to 1.35. The extra .05 is because our interest did work along the way. (We’re assuming the growth happens in whatever time is necessary, then stops afterwards.)

@vivek: Practice problems are key, there’s some at Khan Academy or http://www.exampleproblems.com/.

@chris: Great question. When there’s multiple terms it helps to break them apart:

$$e^{-(ax + bx^2)} = e^{-ax} \cdot e^{-bx^2}$$

Basically, we have two decay factors acting simultaneously (they are decay factors, not growth factors, since the exponent is negative, and will be shrinking our value).

The first is a regular decaying exponential function, e^{-ax}. The second is an exponential function which is shrinking extremely quickly – with the square of time.

The general term e^{-bx^2} is a Gaussian function (bell curve) [http://en.wikipedia.org/wiki/Gaussian_function] and is basically a decaying exponential on steroids. Instead of a nice sloping curve it will be shrinking extremely quickly. What a regular exponential would see in 100 time periods, it sees in 10. I’m not that familiar with this function (or stats in general) but would be a fun thing to investigate later on.

thanks so much Kalid, the tip off that it is a bell curve is great . and the description of it as a decaying exponential on steroids really helps :slight_smile: i will dig further in to the function e^(-bx^2).

hmm, I find it a little odd that it is used to describe a physical phenomenon, killing of cells with radiation, and that the good old regular decay is not sufficient.

Awesome, glad it helped. Yep, I’m not sure why it needs both: maybe there is a bell curve of possibilities, each of which is diminishing exponentially?

Thank you! Finally I can begin to grasp the weird e thing! I have always enjoyed mathematics, but everytime I encountered e I became frustrated. Your e-xplanation is exactly what I needed!

Hello Kalid,

Came here after finishing Barbara Oakley’s “Learning How to Learn” course on coursera. I am thankful to her that not only she has designed a perfect course but she has introduced many learning gems such as your website in the course.

I was always struggling to understand mathematical terms and its use in real life. From this first article I am very sure that I will finally find peace with Maths.

Khalid,
Wow , this is THE article I was looking for. I am now relearning math and its history as a hobby. Math books always give vague abstract ideas and trying visualize is difficult. And this article is really helpful.

It would be nice if you put together, history part of it. Like

1.What made humans to look for this solution?
2.How did they figure out they can use this number as a base and build math around it.
3.Some engineering equations as examples, which uses e and how to read it in plain english.

Thanks!

good job

Thanks .I am 77 years age. had been searching the meaning of e. Makes real sense. Also it is an importand input to calculus in differnciation e powered x .
Radio active decay growth of Bacterial oxygen demand and growth of crystals are all so many to explain e.
MMM

Hi,

I came across an example of a financial calculation. I think it is definitely connected with the ideas of growth presented here. Still, I cannot figure out how that fraction in the last computation was constructed. Any ideas?

Thanks,
Jiri

— Example —
loan = $280,000

interest rate = 3.7% / year
term = 30 years
n = term * 12/year => 360
r = interest rate / (12 / year) => 0.00308

monthly payment = r/(1-(1+r)^(-n)) * loan = $1,288.79

Hello ,

One very important exponential equation is the compound-interest formula:

A=P(1+r/n)^nt
…where “A” is the ending amount, “P” is the beginning amount (or “principal”), “r” is the interest rate (expressed as a decimal), “n” is the number of compounding a year, and “t” is the total number of years.

In General if interest rate is expressed in year (Annual interest rate) , t" must be expressed in years, because interest rates are expressed that way. If an exercise states that the principal was invested for six months, you would need to convert this to 6/12 = 0.5 years; if it was invested for 15months, then t = 15/12 = 1.25 years; if it was invested for 90 days, then t = 90/365 of a year; and so on ………

In example 1: Growing crystal:

Rate is expressed in 24 hours (100% every 24 hours). Should “t” in this case expresse in hours?
As you showed in example 1, you express t in years……

Best regards,

Amazing.

Healing’s Dragon

to discover difficulties to improve my web page!I suppose its alright to help make utilization of a number of of the ideas!!

Absolutely fantastic !!! First ever such mind-blowing intuitive article seen by me on the internet.

I was very very eager to search for what does it actually happen when continuous compounding takes place. It is easy to understand the formula for non-continuous compounding. But I wanted the same logical explanation for usage of ‘e’ as well.

Thanks a lot Kalid !!! You are among those teachers who are the urgent necessity of the academic fraternity.

What an article. 3 words : Excellent. Awesome. Superb.

Thank you, great explanation.